Ghaleb Gumah

Al-Balqa Applied University, Faculty of Engineering Technology, Faculty Member

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Papers by Ghaleb Gumah

Application of reproducing kernel Hilbert space method for solving second-order fuzzy Volterra integro-differential equations

Advances in Difference Equations, Dec 1, 2018

In this article, we propose a new method that determines an efficient numerical procedure for sol... more In this article, we propose a new method that determines an efficient numerical procedure for solving second-order fuzzy Volterra integro-differential equations in a Hilbert space. This method illustrates the ability of the reproducing kernel concept of the Hilbert space to approximate the solutions of second-order fuzzy Volterra integro-differential equations. Additionally, we discuss and derive the exact and approximate solutions in the form of Fourier series with effortlessly computable terms in the reproducing kernel Hilbert space W 3 2 [a, b] ⊕ W .3 2 [a, b]. The convergence of the method is proven and its exactness is illustrated by three numerical examples.

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Research Article On the Homotopy Analysis Method for Fractional SEIR Epidemic Model

Abstract: This study investigates the accuracy of solution for fractional-order an SEIR epidemic ... more Abstract: This study investigates the accuracy of solution for fractional-order an SEIR epidemic model by using the homotopy analysis method. The homotopy analysis method provides us with a simple way to adjust and control the convergence region of the series solution by introducing an auxiliary parameter. Mathematical modeling of the problem leads to a system of nonlinear fractional differential equations. Indeed, we find the analytical solution of the proposed model by Homotopy analysis method which is one of the best methods for finding the solution of the nonlinear problem. Numerical simulations are given to illustrate the validity of the proposed results.

Numerical solutions of special fuzzy partial differential equations in a reproducing kernel Hilbert space

Computational and Applied Mathematics, 2022

On the nonautonomous Belousov–Zhabotinsky (B–Z) reaction

Rendiconti del Circolo Matematico di Palermo Series 2, 2022

A Reliable Computational Method for Solving First-order Periodic BVPs of Fredholm Integro-differential Equations

The object of this paper is to develop a novel adaptive reproducing kernel Hilbert space method (... more The object of this paper is to develop a novel adaptive reproducing kernel Hilbert space method (RKHSM) to solve a class of first-order periodic Fredholm integro-differential equations. The analytical and approximate solution is represented in the form of series in the reproducing kernel space. The n-term approximation is obtained and proved to converge to the analytical solution. Furthermore, the solution methodology is based on generating orthogonal basis from the obtained kernel functions; whilst the orthonormal basis is constructing in order to formulate and utilize the solutions with its series form. Finally, efficacious computational experiments are reported to illustrate the accuracy, generality, superiority and applicability of the presented method.

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On the stability of a class of slowly varying systems

Journal of Inequalities and Applications, 2018

Slowly varying systems are common in physics and control engineering and thus stability analysis ... more Slowly varying systems are common in physics and control engineering and thus stability analysis for those systems has drawn considerable attention in the literature. This paper uses the "frozen time approach" to derive Lyapunov inequality conditions for the stability of a wide class of slowly varying systems. These conditions refine those developed in (Khalil in Nonlinear Systems, 2002) and display generality and effectiveness for both linear and nonlinear systems. To illustrate the utility of the proposed results, an example has been included.

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On the Homotopy Analysis Method for Fractional SEIR Epidemic Model

Research Journal of Applied Sciences, Engineering and Technology, 2014

This study investigates the accuracy of solution for fractional-order an SEIR epidemic model by u... more This study investigates the accuracy of solution for fractional-order an SEIR epidemic model by using the homotopy analysis method. The homotopy analysis method provides us with a simple way to adjust and control the convergence region of the series solution by introducing an auxiliary parameter. Mathematical modeling of the problem leads to a system of nonlinear fractional differential equations. Indeed, we find the analytical solution of the proposed model by Homotopy analysis method which is one of the best methods for finding the solution of the nonlinear problem. Numerical simulations are given to illustrate the validity of the proposed results.

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An Expansion Iterative Technique for Handling Fractional Differential Equations Using Fractional Power Series Scheme

Abstract: In this study, we present a new analytical numerical technique for solving a class of t... more Abstract: In this study, we present a new analytical numerical technique for solving a class of time Fractional Differential Equations (FDEs) with variable coefficients based on the generalized Taylor series formula in the Caputo sense. This method provided the solution in the form of a rapidly convergent power series under a multiple fractional differentiability with easily computable components. An efficacious experiment is given to guarantee the procedure, to illustrate the theoretical statements of the present technique and to show its potentiality, generality and superiority for solving wide range of FDEs. The results reveal that the method is easy to implement, very effective, fully compatible with the complexity of such problems, straightforward and simple.

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Solutions to Uncertain Volterra Integral Equations by Fitted Reproducing Kernel Hilbert Space Method

We present an efficient modern strategy for solving some well-known classes of uncertain integral... more We present an efficient modern strategy for solving some well-known classes of uncertain integral equations arising in engineering and physics fields. The solution methodology is based on generating an orthogonal basis upon the obtained kernel function in the Hilbert space 1 2 [ , ] in order to formulate the analytical solutions in a rapidly convergent series form in terms of their -cut representation. The approximation solution is expressed by -term summation of reproducing kernel functions and it is convergent to the analytical solution. Our investigations indicate that there is excellent agreement between the numerical results and the RKHS method, which is applied to some computational experiments to demonstrate the validity, performance, and superiority of the method. The present work shows the potential of the RKHS technique in solving such uncertain integral equations.

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$Research paper thumbnail of On the consistency of local fractional semilinear Duhem model$

On the consistency of local fractional semilinear Duhem model

In this paper, we introduce the local fractional semilinear and the local fractional Dahl models ... more In this paper, we introduce the local fractional semilinear and the local fractional Dahl models and investigate their consistency with hysteresis. The notion of consistency was initially introduced in Ikhouane (Math Control Signals Syst 25:294–310) to analyze the hysteretic behavior of causal systems and to describe analytically the so-called “hysteresis loops”. The validity of our results is confirmed by numerical simulations.

Numerical solutions of hybrid fuzzy differential equations in a Hilbert space

The main goal of this work is to study a numerical method for certain hybrid fuzzy differential e... more The main goal of this work is to study a numerical method for certain hybrid fuzzy differential equations with an application of a reproducing kernel Hilbert space technique for fuzzy differential equations. Meanwhile, we construct a system of orthogonal functions of the space W 2 2 [ a , b ] ⊕ W 2 2 [ a , b ] depending on a Gram-Schmidt orthogonalization process to get approximate-analytical solutions of a hybrid fuzzy differential equation. A proof of convergence of this method is also discussed in detail. The exact as well as the approximate solutions are displayed by a series in terms of their α-cut representation form in the Hilbert space W 2 2 [ a , b ] ⊕ W 2 2 [ a , b ] . To demonstrate behavior, efficiency, and appropriateness of the present technique, two different numerical experiments are solved numerically in this paper.

Optimization technique for solving fuzzy partial differential equations under strongly generalized differentiability

In this article, we develop and analyze the use of the combined Laplace transform-homotopy pertur... more In this article, we develop and analyze the use of the combined Laplace transform-homotopy perturbation method C(LT-HPM) to find the exact and approximate solutions for fuzzy partial differential equations under strongly generalized differentiability. The C(LT-HPM) allows the solution of the fuzzy partial differential equation to be calculated in the form of an infinite series in which the components can be easily computed. The method is tested on some linear and nonlinear fuzzy partial differential equations with fuzzy initial conditions to show the effectiveness and accuracy of this method.

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Dynamics of a harvested stage‐structured predator–prey model with distributed maturation delay

Mathematical Methods in the Applied Sciences

$Research paper thumbnail of Estimates of certain paraxial diffraction integral operator and its generalized properties$

Estimates of certain paraxial diffraction integral operator and its generalized properties

Advances in Difference Equations

Soft computing technique for a system of fuzzy volterra integro-differential equations in a Hilbert space

Applied Numerical Mathematics

Numerical Investigation for Solving Two-Point Fuzzy Boundary Value Problems by Reproducing Kernel Approach

by Hammad Khalil and Ghaleb Gumah

Applied Mathematics & Information Sciences, 2016

Solutions to Uncertain Volterra Integral Equations by Fitted Reproducing Kernel Hilbert Space Method

Journal of Function Spaces, 2016

We present an efficient modern strategy for solving some well-known classes of uncertain integral... more We present an efficient modern strategy for solving some well-known classes of uncertain integral equations arising in engineering and physics fields. The solution methodology is based on generating an orthogonal basis upon the obtained kernel function in the Hilbert spaceW21a,bin order to formulate the analytical solutions in a rapidly convergent series form in terms of theirα-cut representation. The approximation solution is expressed byn-term summation of reproducing kernel functions and it is convergent to the analytical solution. Our investigations indicate that there is excellent agreement between the numerical results and the RKHS method, which is applied to some computational experiments to demonstrate the validity, performance, and superiority of the method. The present work shows the potential of the RKHS technique in solving such uncertain integral equations.

Application of Reproducing Kernel Hilbert Spaces to a Minimization Problem with Prescribed Nodes

East Asian Journal on Applied Mathematics, 2011

The theory of reproducing kernel Hilbert spaces is applied to a minimization problem with prescri... more

Solutions to Uncertain Volterra Integral Equations by Fitted Reproducing Kernel Hilbert Space Method

Journal of Function Spaces, 2016

We present an efficient modern strategy for solving some well-known classes of uncertain integral... more We present an efficient modern strategy for solving some well-known classes of uncertain integral equations arising in engineering and physics fields. The solution methodology is based on generating an orthogonal basis upon the obtained kernel function in the Hilbert spaceW21a,bin order to formulate the analytical solutions in a rapidly convergent series form in terms of theirα-cut representation. The approximation solution is expressed byn-term summation of reproducing kernel functions and it is convergent to the analytical solution. Our investigations indicate that there is excellent agreement between the numerical results and the RKHS method, which is applied to some computational experiments to demonstrate the validity, performance, and superiority of the method. The present work shows the potential of the RKHS technique in solving such uncertain integral equations.

A Reliable Computational Method for Solving First-order Periodic BVPs of Fredholm Integro-differential Equations

by Mohammed H. Al-Smadi, Asad Freihat, Ahmad El-Ajou, and Ghaleb Gumah

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